See also the
Dr. Math FAQ:
Browse High School Permutations/Combinations
Stars indicate particularly interesting answers or
good places to begin browsing.
- Distributing Apples [10/11/2002]
Eight identical apples are to be given out to four children. How many
different ways of distributing the apples are there if each child must
receive at least one apple?
- Distributing Objects [08/12/2001]
Is there a formula for distributing n indistinguishable objects into k
- Divisibility by Four [04/19/1997]
How many 6-digit numbers are divisible by 4, if we allow no repeated
- Divisors [11/07/1997]
What kinds of numbers have exactly 3 divisors? Do bigger numbers
necessarily have more divisors? Is there a way to figure out how many
divisors 1,000,000 has without actually listing and counting them?...
- Drawing Marbles [05/05/1998]
A jar contains 19 different marbles. I draw 13 marbles, one at a time,
but replace the marble back before taking the next....
- English Exam Probability [12/17/2001]
Given a list of 25 potential exam questions, with 10 of them on the
actual exam and of which students write essays on 2, how many questions
should be prepared to be 95% confident you won't write on a question you
- Enumeration in Combinatorial Analysis [11/28/1996]
How many 3 digit numbers have the digital sum of nine?
- Examples of the Fundamental Counting Principle [02/17/2001]
There are three ways to go from Town A to Town B, and four ways to go
from Town B to Town C. How many different ways are there to go from Town
A to Town C, passing through Town B?
- Explaining Permutation to Young Students [01/20/2007]
To calculate how many different ways four people can sit in a row we
use 4 * 3 * 2 * 1. Why isn't it 4 + 3 + 2 + 1?
- Fair Chance: Three Things, One Coin to Flip [03/12/2003]
How can I choose with equity (p = 1/3) among 3 things (door, person,
etc.) having only a (perfect) coin and not infinite time? How can I
generate an event with probability p = 1/3 with one perfect coin in a
finite number of tosses?
- Finding Catalan Numbers [12/15/1999]
What are Catalan numbers and what applications do we have for them?
- Finishing a Best-of-Seven Series [06/19/2009]
A best-of-seven series is currently 2 games to 1 in favor of team A.
If the two teams are equally likely to win a given game, what is the
probability that A wins the series by reaching 4 games before B does?
- Four Colors, Eighths of a Circle [10/15/2001]
Divide a circle in eighths. Use 4 colors to color the segments. Colors
may be repeated as long as you use all 4 colors at least once. What are
the total combinations possible?
- Four-Letter Combinations [09/06/99]
How can I find the number of four-letter combinations of the letters in
the words COLLEGE and PRINCIPAL?
- Freezer Roulette [07/26/2002]
Two people are trapped in a small freezer that is slowly getting
colder and colder... Is it possible for them to escape? If so, how?
What’s the maximum number of minutes required to escape given ANY
initial combination of button states?
- Frogs Changing Sides [09/04/2002]
What is the least number of jumps needed for all the frogs to trade
- Full House in Draw Poker [7/28/1996]
A player is dealt 5 cards - a pair of aces and three other unpaired
cards. What is the probability of winding up with a full house...
- Galois Theory and Finite Permutation Groups [06/01/2007]
Do the elements (123456)(78) and (17) generate the symmetric group of
- Generating All Permutations of a Set [05/11/2000]
How can I write out all the permutations of a set of objects in a non-
- Generating Eight-Character Passwords [03/08/2002]
Given some restrictions, calculate the number of possible 8-character
- Generating Function of Catalan Numbers [04/04/2000]
Can you explain the recurrence relation for the Catalan numbers?
- Generating Permutations With a Computer [06/20/2002]
I want to write a computer program to solve 'Jumble' word puzzles.
- Generating Permutations With a Computer [06/20/2002]
I have a friend named REBECCA. How can I generate all the
distinguishable permutations of all the letters of her name?
- Generating Possible Lottery Outcomes [3/29/1996]
What math rule would I need to follow if I wanted to generate all
possible combinations in a 50 number draw 6 lottery?
- Gray Code [02/16/2001]
What is Gray code? How does it relate lists of n-tuples of 0's and 1's to
traversing the corners of an n-dimensional cube?
- Handshake Problem [05/07/1997]
If you have n people in a group and each shakes everyone else's hand only
once, how many handshakes take place?
- Handshake Problem Variant [07/08/1999]
Five couples go to a party and start shaking hands. One of the men
shouts, "Stop! How many hands did you shake?" Every person gives a
- Handshakes at a Party [6/23/1996]
At a party, everybody shakes hands with each other once. There are 45
handshakes. How many people are there at the party?
- Hockey League Tournament Schedule [03/09/2000]
Can you help me construct a schedule for a round robin style junior
hockey tournament with two divisions (one of 6 teams, one of 10), using
only four fields?
- How Many Barbies [08/11/1999]
How many different Barbies can I make if I can choose from 4 skin colors,
3 eye colors, 4 hairstyles, and 6 hair colors?
- How Many Different Boxes of Donuts Can Be Made? [03/01/1998]
Using the "stars and bars" method to count how many different boxes of a
dozen donuts can be made from five different types of donuts.
- How Many Different Dinners Can Be Ordered? [05/25/2007]
In the Arby's "5 for $5.95" dinner promotion, customers choose any 5
of 8 menu items for $5.95. You can get 5 different items, 5 of the
same item, or anything in between. How many different dinners are
- How Many Distinct Patterns? [01/15/2001]
Given a large equilateral triangle divided into four smaller equilateral
triangles, if two edges are painted white and the rest are painted black,
how many distinct patterns are possible?
- How Many Games in the Tournament? [01/15/2002]
There are eight teams in a single-elimination tournament. Each team gets
to play until it loses. How many games will be played in the tournament?
- How Many Handshakes? [01/29/2001]
There are 40 people in a room. They shake each other's hands once and
only once. How many handshakes are there altogether?
- How Many Handshakes? [08/18/2005]
Thirty people at a party shook hands with each other. How many
handshakes were there altogether? Draw a diagram and see if you can
establish a pattern by collecting data in a table for one, two, three,
four and five people shaking hands.
- How Many Possible Pizza Topping Combinations? [12/21/2004]
Dominos offers 13 possible pizza toppings, and each topping can be
ordered in single, double, or triple layers. How many different
possible pizzas can be ordered?
- How Many Regions are Generated? [05/29/2003]
Draw a circle and pick n points on it. Now join every point to every
other point and suppose that the points have been picked so that no
three chords go through one point (i.e. every intersection is the
intersection of exactly two chords).
- Identical/Non-identical Groups; Derangements [03/10/2000]
Formulas for dividing things into identical or different, non- identical
- Induction With Binomial Coefficients [10/16/2000]
Prove that the sum from i = 1 to n of (i+k-1 choose k) equals (n+k choose